Bayesian networks (BN) are a popular class of graphical probabilistic models which are motivated by Bayes theorem. Such networks have been increasingly applied to various computation applications, such as computational biology and computer vision. Current Bayesian network parameter learning approaches can be classified into two categories: frequentist approach with maximum likelihood (ML) estimation, and a Bayesian approach with maximum a posterior (MAP) estimation. ML only uses statistical counts from the data to estimate the parameters in a Bayes net while MAP employs certain type of priori statistical belief to further regulate the statistical counts from the data. Therefore, MAP parameter estimation is a combinatorial result computed from both data and prior knowledge.
Prior knowledge can be loosely defined as any statements involving network structure and parameter properties. For example, a domain expert can provide information on a local structure by specifying conditional independence among a subset of variables or even fully specifying the causality between variables, thus direction of this edge. A domain expert might also provide knowledge on network parameters ranging from directly assigning values to entries in a conditional probability table (CPT) to introducing inequality and equality constraints in one distribution or across multiple conditional distributions. In addition, domain knowledge might also define a prior distribution over the parameters. Moreover, some knowledge imposes equality and inequality constraints over the hyperparameters in the prior distribution function.
Among various domain knowledge, qualitative statements are perhaps the most generic and intuitive information in a domain. These statements describes (in)dependence and causality among domain variables. By definition, “(in)dependence” links multiple entities in a joint representation and “causality” specifies the conditions and direction within this joint set. In Bayesian framework, the (in)dependence and causality in the qualitative domain knowledge defines a set of inequality relation(s) between a child node's probabilities given all possible conFig.s of its parents. Since nearly every BN admits (in)dependence and causality given structure, this type of constraints provide the most generic and useful priori constraints to the parameter learning task. Moreover, it is less demanding for a domain expert to specify an inequality constraint over entries in a CPT than giving specific values on individual entry or defining the hyper-parameters of a Dirichlet prior.
Researchers have proposed a number of algorithms to learn Bayesian network parameters by utilizing various forms of prior knowledge, such as dirichlet function[2], [3]. In certain references [4], [5], [6], parameter learning schemes for various graphical models incorporating parameter sharing constraints is proposed. For example, parameter equality in one multinomial conditional distribution. The forms of the constraints are limited to either parameter sharing or inequality constraints within one conditional distribution, such as P(A|B)>P(A|˜B). More generic and important inequality constraints, such as P(A|B)>P(A|˜B) is not addressed by their methods. In other references [10] and [11], methods are proposed to deal with the inequality constraints in parameter learning.
Thus, the inequality constraints have been proposed and used in qualitative probabilistic inference [18], [19]. However, due to the lacks of quantitative measurements in the constraints, they have long been ignored in incorporating with the quantitative training data in any BN learning process.
The current drug development strategy is primarily focused on developing tight inhibitors for a single target, which may not be the most effective way to treat a disease. The recent advancement of systems biology clearly demonstrated that the genes/proteins are tightly connected to each other. The role of a drug target in the genetic network is no doubt a determinant factor for the efficacy of the inhibitor(s). In addition, considering the redundancy and robustness of the genetic network, effective treatment of a disease may require simultaneous inhibition of multiple proteins. Developing pharmaceutical targeting a set of proteins in a pathway will open a new territory in drug discovery and will be a trend in the near future.
Given the complexity of genetic networks, it is difficult, if not impossible, to manually select a combination of proteins as the drug targets and predict the effects of inhibiting these proteins. Therefore, a computational approach that can tackle this problem in a systematic and unbiased manner is needed for pharmaceutical companies that are constantly searching for new drug targets.